Hi Heather,
It may be difficult to get an answer from the forums about that, as I suspect that your site has either a Magento Extension or some custom coding for gift products. By default, Magento doesn't add products to the cart when you use a coupon code.
Here's an example of a Magento Extension that Does have this capability: https://marketplace.magento.com/amasty-promo.html
My recommendation would be to try to identify if you're using a publicly sold extension, and to open a ticket with the company you purchased the extension from. If this solution was created custom for you, and was not part of an off-the-shelf extension, it would be best to connect with your web developers for more information.
Best of luck!